Quantitative assessment of the universal thermopower in the Hubbard model

As primarily an electronic observable, the room-temperature thermopower S in cuprates provides possibilities for a quantitative assessment of the Hubbard model. Using determinant quantum Monte Carlo, we demonstrate agreement between Hubbard model calculations and experimentally measured room-temperature S across multiple cuprate families, both qualitatively in terms of the doping dependence and quantitatively in terms of magnitude. We observe an upturn in S with decreasing temperatures, which possesses a slope comparable to that observed experimentally in cuprates. From our calculations, the doping at which S changes sign occurs in close proximity to a vanishing temperature dependence of the chemical potential at fixed density. Our results emphasize the importance of interaction effects in the systematic assessment of the thermopower S in cuprates.

Statistical error bars denoting ±1 standard error of the mean are shown for all measurements, except for Supplementary Fig. 2 that has none.Error bars are determined by bootstrap resampling (100 bootstraps) [1], except for error bars determined by jackknife resampling [2]: n in the inset of Fig. 2b in the main text, S Kelvin in Supplementary Fig. 3, and 16 × 16 S Kelvin data in Supplementary Fig. 7b.Simulation cluster size is 8 × 8 for all results, unless otherwise specified.The maximum imaginary time Trotter discretization is dτ = 0.02/t in the chemical potential tuning process, and dτ = 0.05/t for other thermodynamic and transport measurements, unless otherwise specified.At high temperatures, the smallest number of imaginary-time slices used in the Trotter decomposition is L = β/dτ = 20.For MaxEnt analytic continuation, we choose the model function by using the same hightemperature annealing procedure as in Ref. [3], except for Supplementary Fig. 1.We determine spectra in the infinitetemperature-limit, using a moments expansion method, which serves as the model function at the highest temperature, except for Supplementary Fig. 2, similar as in Refs.[3][4][5].To determine the adjustable parameter which assigns weights of statistics and entropy in the maximized function in MaxEnt, we use the method of Ref. [6].Other details in methods and parameter choices are mostly the same as Ref. [3].

Supplementary Note 2: Formalism
We set ℏ to 1 throughout the paper.We consider the response due to a temperature gradient ∇T and electric field E = −∇V .We define µ = µ+e * V so that ∇µ = ∇µ−e * E, where charge e * = −e for electrons.The responses along the x direction in terms of DC transport coefficients L O1O2 (ω = 0 value of Eq. ( 6) in the main text) are [5,7] The thermopower S is defined as giving us Eq. ( 5) in the main text.In Supplementary Eq. ( 3), we used Onsager's reciprocity relations [8] Setting Z = Tr(e −β(H−µN ) ) as the partition function, from Eq. ( 6) in the main text, In the case of (6) where |i i ⟩ (E i1 ) are eigenstates (eigenvalues) of the grandcanonical Hamiltonian H −µN .From Supplementary Eq. ( 6) we obtain Re L OO (ω) = Re L OO (−ω).By Kramers-Kronig relations, Im L OO (ω = 0) = 0.
According to Eq. ( 6) in the main text, we may write where λ sub is an arbitrary non-zero real constant [9].With Supplementary Eq. ( 9), Supplementary Eq. ( 8) can be gener-alized, Re L OO (ω) is guaranteed to be positive definite in Supplementary Eq. ( 6) when O 1 = O 2 = O, in which case Max-Ent analytic continuation is applicable.However, in calculation of the thermopower, Re L J Q,x Jx (ω) + Re L JxJ Q,x (ω) can change its sign as a function of ω, so it cannot be directly calculated from ⟨T τ J Q,x (τ )J x ⟩ + ⟨T τ J x (τ )J Q,x ⟩ using Supplementary Eq. ( 10) through MaxEnt.So, according to Supplementary Eqs. ( 4) and ( 9), we calculate L J Q,x Jx using Since Jx is also real.In principle, if there are no errors in every L OO term on the right hand side of Supplementary Eq. ( 11), then the result of L J Q,x Jx from Supplementary Eq. ( 11) is λ sub independent.However, systematic errors introduced by the analytic continuation process propagate in the calculation of Supplementary Eq. ( 11), which is reflected by S exhibiting some degree of λ sub dependence.In Supplementary Fig. 1, we show S as a function of λ sub for four sets of parameters as examples.As long as |λ sub | ≳ 1, the λ sub dependence is relatively weak.Therefore, as a reasonable choice, we use λ sub = 2 in this work.

Supplementary Note 3: Lifshitz transition
We calculate the density of states (DoS) from the DQMC results of the local Green's function G using MaxEnt analytic continuation.For the model function in MaxEnt, we start with using the flat model at the highest temperature k B T /t = 8, and proceed with lower temperatures using the high-temperature annealing procedure.In Supplementary Fig. 2 we show doping dependence of DoS(ω) for fixed U/t = 6, t ′ /t = −0.25, and k B T /t = 0.25.We observe that the Lifshitz transition, at which the quasiparticle peak crosses the Fermi level at ω = 0, happens at doping p ∼ 0.26, which is much higher than the sign change doping of S at p ∼ 0.15 in Fig. 1 in the main text for the corresponding parameter set.Therefore, the sign change doping of S is not associated with the Lifshitz transition.

Supplementary Note 4: Kelvin formula
The Kelvin formula for thermopower is [11] where s is the entropy density and n is the particle density.
To obtain the second equality in Supplementary Eq. ( 13), we consider the thermodynamic potential density f = ϵ − sT − µn, where ϵ is the energy density.Using the first law of thermodynamics, we obtain d(f + µn) = −s dT + µ dn.Equating then gives us the Maxwell relation leading to the second equality in Supplementary Eq. ( 13).
The specific heat (considering Supplementary Eq. ( 14)) is So from Supplementary Eqs. ( 13) and (20), we obtain Therefore, the temperature dependence of S Kelvin is directly related to doping dependence of the specific heat c v .In the 0.5 0.0 0.5  3. Comparison between Hubbard model (solid lines, obtained by DQMC) and atomic-limit (dashed lines, from Supplementary Eq. ( 25)) results of S Kelvin for large interactions: (a) U/t = 16, t ′ /t = 0, (b) U/t = 12, t ′ /t = 0, (c) U/t = 16, t ′ /t = −0.25, and (d) U/t = 12, t ′ /t = −0.25.For these parameters, the maximum dτ for chemical potential tuning is 0.01/t.Error bars denote ±1 standard error of the mean determined by jackknife resampling.main text, we use doping p = 1 − n instead of n.So we rewrite Supplementary Eq. (21) as For the calculation of −∂ 2 s/(∂p∂T ) in Fig. 4 in the main text, to rule out data points with large error bars in the spline fitting process, for the fitting of c v , the lowest temperature considered is k B T = t/3.5 for U/t = 6 and k B T = t/3 for U/t = 8; for S Kelvin , the lowest temperature in the fitting range is k B T = t/4.5 for U/t = 6 and k B T = t/3.5 for U/t = 8.Since the measurements of c v involve energy fluctuation and therefore contains correlators with up to 8 fermion operators, while S Kelvin contains up to 6, for the same set of parameters, c v data generally has larger statistical error than S Kelvin .Therefore a higher lowest temperature is chosen for fitting c v than that for S Kelvin .

Supplementary Note 5: Atomic limit
In this note we derive the atomic-limit (t, t ′ ≪ k B T, U ) approximation of S and S Kelvin .
Considering the condition t, t ′ ≪ U , we divide the Hamiltonian of Eq. ( 1) in the main text into the interaction part H 0 ∝ U as the unperturbed Hamiltonian and the kinetic part ∆H as the perturbative term.Namely, where ∆H(τ 1 ) = e τ1(H0−µN ) ∆He −τ1(H0−µN ) , the Using Supplementary Eq. ( 24) evaluated under the occupation basis (the eigenstates of H 0 ), Eq. ( 2) in the main text can be obtained to leading order.This leads to the atomic-limit approximation In the same limit, we can calculate the average density ⟨n⟩.Applying Supplementary Eq. ( 23), we find Therefore, to leading order, ⟨n⟩ = 2e which allows us to determine µ for any given density n in the atomic limit.In Supplementary Fig. 3, we compare S Kelvin calculated using DQMC with the atomic-limit approximation of S Kelvin , Supplementary Eq. (25).Large interactions U/t = 16 and U/t = 12 are selected.At high temperatures, where the condition t, t ′ ≪ k B T is satisfied, the simulation results match the atomic-limit approximations well.As temperature decreases and this condition breaks down, S Kelvin deviates from its atomic-limit approximation.Now, we derive the atomic-limit approximation for thermopower S. Still using the occupation basis, and replacing O 1 and O 2 with J x or J E,x operators in Supplementary Eq. ( 24), the J E,x − J x and J x − J x correlation functions to leading order are where Z 0 = 1 + 2e βU/2+βµ + e 2βµ .Notice that any term of the form (e −(β−τ )U + e −τ U ) multiplied by a quantity in- through Supplementary Eq. (10).Summing up magnitudes of such terms provides the integrated weights of Re L O1O2 (ω) + Re L O2O1 (ω) around ω = 0. So, using finite-frequency Onsager relations [7], Supplementary Eqs. ( 10), (29), and (30), with |ε| < U , we have Here, both Re L JxJx (ω) and Re L J E,x Jx (ω) are proportional to δ(ω) at low frequencies, so they are both infinite at ω = 0.
To make both Re L JxJx (ω = 0) and Re L J E,x Jx (ω = 0) finite, we introduce a small scattering rate [12][13][14][15] in both terms cancels out when we take their ratio and gives us the ratio of corresponding weights.Under this assumption, combining Supplementary Eqs. ( 3), (31), (32), and that J Q = J E − µJ, we obtain the atomic-limit approximation of thermopower to leading order, S = lim An interesting observation in the atomic limit is that t and t ′ affect S (a transport property) in Supplementary Eq. (33), but not S Kelvin (a thermodynamics property) in Supplementary Eq. ( 25).If we take t ′ = 0, the expression Supplementary Eq. ( 33) is equivalent to corresponding expressions of S derived and discussed in Refs.[12][13][14][15], where the chemical potential is different from our definition by U/2 due to the difference in the Hamiltonian definition.
Such terms do not contribute to the DC values of transport coefficients.Any term independent of τ in ⟨T τ O 1 (τ )O 2 ⟩ + ⟨T τ O 2 (τ )O 1 ⟩ corresponds to a delta function at ω = 0 in Re L O1O2 (ω) + Re L O2O1 (ω).